
\subsubsection{Mathematical model}
Objective:
\begin{equation}
\max{\sum_{i \in I}{u_i}}
\end{equation}

Constraints:
\begin{align}
&0\le P_g \le 1 \\
&\forall_{i \in I}{x_i\ge 0} \\
&\forall_{i \in I}{c_i\ge 0} \\
&\forall_{i \in I}{y_i = x_i + c_i} \\
&\sum_{i \in I}{x_i} = P_g \cdot G \\
&\forall_{i \in I}{u_i = c_i + \alpha_i \cdot P_g \cdot G}
\end{align}

\subsubsection{AMPL model}

\begin{lstlisting}[language=c,style=outcode,caption=mill.model.ampl]
################# PARAMETERS
param N;                     # number of the agents
param G;                     # cost of the good
param alpha{1..10} >=0, <=1; # MRS
param y{1..10} >=0;          # endowment
################# VARIABLES
var x{1..N} >=0;             # contribution/tax
var u{1..N};                 # utility
var c{1..N} >=0;             # private constumption
var Pg >= 0, <=1;            # scale of good provision
################# CONSTRAINTS
subject to util {i in 1..N}:
	u[i] = c[i] + alpha[i] * Pg * G;
subject to provide_good:
	sum{i in 1..N} x[i] = Pg * G;
subject to budgets {i in 1..N}:
	y[i] = x[i] + c[i];
################# OBJECTIVE
maximize mill:
	sum{i in 1..N} u[i];
\end{lstlisting}

\subsubsection{Results}
\begin{lstlisting}[language=c,style=outcode,caption=Utilitarian results for scenario A]
DONLP2: Success! KKT conditions satisfied
F = 1446.023901
8 iters; 13 function, 4 gradient evals
904 component constraint, 21 grad evals
final objective scaling = 31.6386
norm(grad(f)) = 3.16228, Lagrangian violation = 1.15065e-13
feas. violation = 7.81597e-14, dual feas. violation = 0
###### MILL ###### Pg = 1.00 ###
\end{lstlisting}


\begin{table}[H]
\caption{Utilitarian results for scenario A}
\begin{center}
\begin{tabular}{|r||r|r|r|r|r|}
\hline
$i$ & $\alpha_i$ & $y_i$ & $x_i$ & $c_i$ & $u_i$ \\
\hline 
 1& 0.55& 91.15 &9.98 &81.17   &135.87 \\
 2& 0.75& 121.29& 16.81& 104.47& 179.66 \\
 3& 0.54& 100.49& 9.77 &90.72  & 144.78 \\
 4& 0.23& 93.48 &0.00 &93.48   &116.91 \\
 5& 0.43& 107.64& 6.04 &101.60 & 144.47 \\
 6& 0.56& 127.22& 10.38& 116.84& 172.72 \\
 7& 0.72& 98.34 &15.76 &82.57  & 154.60 \\
 8& 0.67& 74.08 &14.17 &59.90  & 127.17 \\
 9& 0.47& 85.57 &7.35 &78.23   &125.01 \\
10& 0.54& 100.62& 9.73 &90.89  & 144.83 \\
\hline
\end{tabular}
\end{center}
\label{tab:mill_res_A}
\end{table}

